Is it possible to model this padding with "inverted snake scale"-like pattern?
Here different alternative with is a more parametric approach using modifiers.



Use a technique similar to Moonboots answer to start your shape.
Add a plane, subdivide-smooth a few times, poke faces, remove all orthogonal edges so you end up with the lozenges pattern.

Select all diagonal edges and with Ctrl + E Edges menu mark them as Sharp.

Now poke you faces again, and select all face-center vertex. You can do this easily by selecting all Sharp edges from the above step with Shift + G > Sharpness, then switch to Vertex mode and invert the selection.

Create a new vertex group and assign to it all the face center vertex selected previously, leaving all other out.



You can now add a new Displace modifier to your mesh and limit it to the above created vertex group. This will parametrically control the scales concavity with the displace amount.

Add an Edge Split modifier and tick only the option Sharp Edges to separate each individual scale. Follow it with a Smooth modifier to non destructively control the gap between scales.

Add a Solidify modifier to add thickness, then follow it by a Subdivide modifier set to Simple. This will add some gometry to "hold the square shape" of the scales.

After that you can add another Subdivide modifier set to Catmull-Clark to add roundness.